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The Korteweg–de Vries equation on the interval
Journal of Mathematical Physics, 2010The initial-boundary value problem for the Korteweg–de Vries equation posed on a finite interval of the spatial variable is considered. Using the method of simultaneous spectral analysis of the associated Lax pair, this problem is mapped into a Riemann–Hilbert problem formulated in the complex plane of the spectral parameter, but with explicit ...
Hitzazis, Iasonas, Tsoubelis, Dimitri
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2009
In this chapter we study the local well-posedness (LWP) for the initial value problem (IVP) associated to the generalized KdV equation. We discuss the local theory for the KdV equation, the modified KdV equation, and the generalized KdV equations. We also show the sharpness of some of these results.
Felipe Linares, Gustavo Ponce
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In this chapter we study the local well-posedness (LWP) for the initial value problem (IVP) associated to the generalized KdV equation. We discuss the local theory for the KdV equation, the modified KdV equation, and the generalized KdV equations. We also show the sharpness of some of these results.
Felipe Linares, Gustavo Ponce
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Journal of Mathematical Physics, 1969
The Korteweg-de Vries equation and the Burgers equation are derived for a wide class of nonlinear Galilean-invariant systems under the weak-nonlinearity and long-wavelength approximations. The former equation is shown to be a limiting form for nonlinear dispersive systems while the latter is a limiting form for nonlinear dissipative systems.
Su, C.-H., Gardner, Clifford S.
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The Korteweg-de Vries equation and the Burgers equation are derived for a wide class of nonlinear Galilean-invariant systems under the weak-nonlinearity and long-wavelength approximations. The former equation is shown to be a limiting form for nonlinear dispersive systems while the latter is a limiting form for nonlinear dissipative systems.
Su, C.-H., Gardner, Clifford S.
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The Korteweg-de Vries equation and beyond
Acta Applicandae Mathematicae, 1995The author reviews a new method for linearizing the initial-boundary value problem of the Korteweg-de Vries (KdV) equation on the semi-infinite line for decaying initial and boundary data. The author also presents a novel class of physically important integrable equations.
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Computability of Solutions of the Korteweg-de Vries Equation
MLQ, 2001zbMATH Open Web Interface contents unavailable due to conflicting licenses.
William Gay +2 more
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A remark on the korteweg-de vries equation
Lettere al Nuovo Cimento, 1984Let u(x, t) be a solution of the KdV equation ut — uxxx— 6uxu = 0, z(x,t) be a solution of the first-order PDE zt- 2uzx = 0 and v(z) be defined in terms of u and z by the formula r[z(x,i),t] = [zx(x,t)]-2 u(x,t) — [zx(x,t)]-1/2[zx{x,t)-1/2}xx. Then vt(z,t) = 0, namely v(z,t) does not change with time (for fixed z).
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Journal of Mathematical Physics, 1971
It is shown that if a function of x and t satisfies the Korteweg-de Vries equation and is periodic in x, then its Fourier components satisfy a Hamiltonian system of ordinary differential equations. The associated Poisson bracket is a bilinear antisymmetric operator on functionals.
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It is shown that if a function of x and t satisfies the Korteweg-de Vries equation and is periodic in x, then its Fourier components satisfy a Hamiltonian system of ordinary differential equations. The associated Poisson bracket is a bilinear antisymmetric operator on functionals.
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The Korteweg-de Vries Equation
1998The Korteweg-de Vries equation, $$ {u_t} + u{u_x} + {u_{xxx}} = 0 $$ (3.1.1) is the simplest equation that includes both the effects of nonlinearity and dispersion. The equation appears in various forms in the literature, sometimes with a factor of 6 or -6 in front of the nonlinear term.
Carlo Cercignani, David H. Sattinger
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Primitive solutions of the Korteweg–de Vries equation
Theoretical and Mathematical Physics, 2020The authors of this paper survey recent results concerning a new family of solutions, called ``primitive solutions'' of the Korteweg-de Vries (KdV) equation, \[ u_t=6uu_x-u_{xxx}. \] In this differential equation \(u=u(x,t)\) is the unknown function. It is the first equation of an infinite sequence of commuting equations called the KdV hierarchy.
Dyachenko, S. A. +3 more
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New exact solutions for Korteweg-de Vries Burgers equation and Korteweg-de Vries equation
MSIE 2011, 2011To obtain new exact solutions for Korteweg-de Vries Burgers equation and Korteweg-de Vries equation, with the aid of symbolic computation, the Korteweg-de Vries Burgers equation and Korteweg-de Vries equation are investigated by using the trigonometric function transform method.
DongBo Cao, null LiuXian Pan, JiaRen Yan
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